Home | | Maths 12th Std | Introduction to Complex Numbers

Chapter: 12th Mathematics : UNIT 2 : Complex Numbers

Introduction to Complex Numbers

when we square a real number it is impossible to get a negative real number.

Introduction to Complex Numbers

Before introducing complex numbers, let us try to answer the question “Whether there exists a real number whose square is negative?” Let’s look at simple examples to get the answer for it. Consider the equations 1 and 2.


This is because, when we square a real number it is impossible to get a negative real number.

If equation 2 has solutions, then we must create an imaginary number as a square root of 1. This imaginary unit √1 is denoted by i .The imaginary number i tells us that i2 = 1. We can use this fact to find other powers of i

 

Powers of imaginary unit i


We note that, for any integer n , in  has only four possible values:  they correspond to values of n when divided by 4 leave the remainders 0, 1, 2, and 3.That is when the integer n 4  or  n 4 , using division algorithm, n can be written as n = 4q + k, 0 k < 4, k and q are integers and we write

 i)n = (i)4q+k = (i)4q(i)k = ((i)4)q(i)k =  (i) q(i)k =(i)k

 

Example 2.1

Simplify the following 

(i) i7 (ii) i1729 (iii) i −1924 + i2018 (iv)  (v) i i2 i3 ….. i40 

Solution

(i) (i)7 = (i)4+3 = (i)3 = -i;

(ii) i1729 = i1729i1 = i

(iii) (i)-1924 + (i)2018 = (i)-1924+0 + (i)2016+2 = (i)0 + (i)2 = 1-1 = 0

(iv)  = (i1 + i2 + i3 + i4) + (i5 + i6 + i7 + i8 ) + … + (i97 + i98 + i99 + i100 +) + i101 + i102

= (i1 + i2 + i3 + i4) + (i1 + i2 + i3 + i4) + … + (i1 + i2 + i3 + i4) + i1 + i2

= {i+(-1)+(-i)+1} + {i+(-1)+(-i)+1} + …. … + {i+(-1)+(-i)+1} + i + (-1)

= 0 + 0 + … 0 + i -1

= −1 + i (What is this number?)

(v) i i2 i3 … i40 = i1+2+3+…+40 = i [40x41]/2 = i820 = i0 = 1

Note

(i) √[ab] = √a√b valid only if at least one of a, b is non-negative.

For example, 6 = √36 = √[(-4)(-9)] = √(-4) √ (-9) = (2i) (3i) = 6i2 = -6, a contradiction.

Since we have taken √[(-4)(-9)] = √(-4) √ (-9), we arrived at a contradiction. 

Therefore √[ab] = √a √b valid only if at least one of a b, is non-negative.

(ii) For y R , y2 ≥ 0


 iy = yi.

 

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
12th Mathematics : UNIT 2 : Complex Numbers : Introduction to Complex Numbers |

Related Topics

12th Mathematics : UNIT 2 : Complex Numbers


Privacy Policy, Terms and Conditions, DMCA Policy and Compliant

Copyright © 2018-2023 BrainKart.com; All Rights Reserved. Developed by Therithal info, Chennai.